An uncountable countable set
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From: Tony Orlow on 28 Oct 2006 14:47 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: >>>>>>>>> You are mentioning balls and time and a vase. But, what I'm asking is >>>>>>>>> completely separate from that. I'm just asking about a math problem. >>>>>>>>> Please just consider the following mathematical definitions and >>>>>>>>> completely ignore that they may or may not be relevant/related/similar >>>>>>>>> to the vase and balls problem: >>>>>>>>> >>>>>>>>> -------------------------- >>>>>>>>> For n = 1,2,..., let >>>>>>>>> >>>>>>>>> A_n = -1/floor((n+9)/10), >>>>>>>>> R_n = -1/n. >>>>>>>>> >>>>>>>>> For n = 1,2,..., define a function B_n: R -> R by >>>>>>>>> >>>>>>>>> B_n(t) = 1 if A_n <= t < R_n, >>>>>>>>> 0 if t < A_n or t >= R_n. >>>>>>>>> >>>>>>>>> Let V(t) = sum_n B_n(t). >>>>>>>>> -------------------------- >>>>>>>>> >>>>>>>>> Just looking at these definitions of sequences and functions from R (the >>>>>>>>> real numbers) to R, and assuming that the sum is defined as it would be >>>>>>>>> in a Freshman Calculus class, are you saying that V(0) is not equal to >>>>>>>>> 0? >>>>>>>> On the surface, you math appears correct, but that doesn't mend the >>>>>>>> obvious contradiction in having an event occur in a time continuum >>>>>>>> without occupying at least one moment. It doesn't explain how a >>>>>>>> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >>>>>>>> that all finite naturals are removed by noon. I never disagreed with >>>>>>>> that. However, to actually reach noon requires infinite naturals. Sure, >>>>>>>> if V is defined as the sum of all finite balls, V(0)=0. But, I've >>>>>>>> already said that, several times, haven't I? Isn't that an answer to >>>>>>>> your question? >>>>>>> I think it is an answer. Just to be sure, please confirm that you agree >>>>>>> that, with the definitions above, V(0) = 0. Is that correct? >>>>>> Sure, all finite balls are gone at noon. >>>>> Please note that there are no balls or time in the above mathematics >>>>> problem. However, I'll take your "Sure" as agreement that V(0) = 0. >>>> Okay. >>>> >>>>> Let me ask you a question about this mathematics problem. Please answer >>>>> without using the words "balls", "vase", "time", or "noon" (since these >>>>> words do not occur in the problem). >>>> I'll try. >>>> >>>>> First some discussion: For each n, B_n(0) = 0 and B_n is continuous at >>>>> zero. >>>> What??? How do you conclude that anything besides time is continuous at >>>> 0, where yo have an ordinal discontinuity???? Please explain. >>> I thought we agreed above to not use the word "time" in discussing this >>> mathematics problem? >> If that's what you want, then why don't you remove 't' from all of your >> equations? > > It is just a letter. It stands for a real number. Would you prefer "x"? > I'll switch to "x". > It is still related to n in such a way that x<0. >>> As for your question, let's look at B_2 (the argument is similar for the >>> other B_n). >>> >>> B_2(t) = 1 if A_2 <= t < R_2, >>> 0 if t < A_2 or t >= R_2. >>> >>> Now, A_2 = -1 and R_2 = -1/2. So, >>> >>> B_2(t) = 1 if -1 <= t < -1/2, >>> 0 if t < -1 or t >= -1/2. >>> >>> In particular, B_2(t) = 0 for t >= -1/2. So, the value of B_2 at zero is >>> zero and the limit as we approach zero is zero. So, B_2 is continuous at >>> zero. >> Oh. For each ball, nothing is happening at 0 and B_n(0)=0. That's for >> each finite ball that one can specify. > > I thought we agreed to not use the word "ball" in discussing this > mathematics problem? Do you want me to change the letter "B" to a > different letter, too? > Call it an element or a ball. I don't care. It doesn't matter. >> However, lim(t->0: sum(B_n| B_n(t)=1))=oo. Why do you conveniently >> forget that fact? > > Your notation is nonstandard, so I'm not sure what you mean. Do you mean > to write > > lim_{x -> 0-} sum_n B_n(x) = oo > > ? If so, I don't understand why you think I've forgotten this fact. If > you look in my previous post (or below), you will see that I wrote, > "Now, V is the sum of the B_n. As t approaches zero from the left, V(t) > grows without bound. In fact, given any large number M, there is an e < > 0 such that for e < t < 0, V(t) > M." > Then don't you see a contradiction in the limit at that point being oo, the value being 0, and there being no event to cause that change? I do. >>>>> In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for >>>>> e < t <= 0. > >>>> There is no e<0 such that e<t and B_n(t)=0. That's simply false. >>> Let's look at B_2 again. We can take e = -1/2. Then B_2(t) = 0 for e < t >>> <= 0. Similarly, for any other given B_n, we can find an e that does >>> what I wrote. >> Yes, okay, I misread that. Sorry. For each ball B_n that's true. For the >> sum of balls n such that B_n(t)=1, it diverges as t->0. >> >>>>> In other words, B_n is not changing near zero. >>>> Infinitely more quickly but not. That's logical. And wrong. >>> Not sure what you mean. >> The sum increases without bound. >> >>>>> Now, V is the >>>>> sum of the B_n. As t approaches zero from the left, V(t) grows without >>>>> bound. In fact, given any large number M, there is an e < 0 such that >>>>> for e < t < 0, V(t) > M. We also have that V(0) = 0 (as you agreed). >>>>> >>>>> Now the question: How do you explain the fact that V(t) goes from being >>>>> very large for t a little less than zero to being zero when t equals >>>>> zero even though none of the B_n are changing near zero? >>>> I'll consider answering that when you correct the errors above. Sorry. > > I believe we now agree that what I wrote is correct. So, let me repeat > my question: > > How do you explain the fact that V(x) goes from being very large for x a > little less than zero to being zero when x equals zero even though none > of the functions B_n are changing near zero? > How do I account for it? I don't, because I don't believe it's true. I "account" for it by saying there is a logical flaw in the argument that says so. That's what I've been saying all along. Cardinality is not a fine eno
From: Tony Orlow on 28 Oct 2006 14:48 David Marcus wrote: > imaginatorium(a)despammed.com wrote: >> Anyway, I'm getting a giggle from hearing about you "reading" Robinson; >> makes me wonder if that's how you can find anything of merit in >> Lester's endless drivel - you just cruise through looking for an >> attractive sentence here or there? > > If you don't realize that the words are supposed to convey rigorous > mathematics, you can read a math book the same way that you do a novel. > I fear that most undergraduates who are not math majors read their math > books this way. > Those poor, lowly undergraduates....
From: imaginatorium on 28 Oct 2006 14:49 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > Tony Orlow <tony(a)lightlink.com> wrote: > >> stephen(a)nomail.com wrote: > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> stephen(a)nomail.com wrote: > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> stephen(a)nomail.com wrote: > >>>>>>>>> What are you talking about? I defined two sets. There are no > >>>>>>>>> balls or vases. There are simply the two sets > >>>>>>>>> > >>>>>>>>> IN = { n | -1/(2^floor(n/10)) < 0 } > >>>>>>>>> OUT = { n | -1/(2^n) < 0 } > >>>>>>>> For each n e N, IN(n)=10*OUT(n). > >>>>>>> Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and "OUT > >>>>>>> (n)". So, you seem to be answering a question he didn't ask. Given > >>>>>>> Stephen's definitions of IN and OUT, is IN = OUT? > >>>>>>> > >>>>>> Yes, all elements are the same n, which are finite n. There is a simple > >>>>>> bijection. But, as in all infinite bijections, the formulaic > >>>>>> relationship between the sets is lost. > >>>>> What "formulaic relationship"? There are two sets. The members > >>>>> of each set are identified by a predicate. > >>>> OOoooOOoooohhhh a predicate! > >>> This is a non answer. > >>> > > > >> That's because it followed a non question. :) > > > > How is "formulaic relationship?" a non question? I do not know > > what you mean by that phrase, so I asked a question about. > > Presumably you do know what it means, but your refusal to > > answer suggests otherwise. > > > > > >>>> If an element satifies > >>>>> the predicate, it is in the set. If it does not, it is not in > >>>>> the set. > >>>>> > >>>> Ever heard of algebra or formulas? Ever seen a mapping between two sets > >>>> of numbers? > >>> This is a lame insult and irrelevant comment. It says nothing > >>> about what a "forumulaic relationship" between sets is. > >>> > > > >> What is there to say? You know what a formula is. > > > > Yes, but I do not know what a "formulaic relationship" is. > > > >>>>> I could define "different" sets with different predicates. > >>>>> For example, > >>>>> A = { n | 1+n > 0 } > >>>>> B = { n | 2*n >= n } > >>>>> C = { n | sin(n*pi)=0 } > >>>>> Are these sets "formulaically related"? Assuming that n is > >>>>> restricted to non-negative integers, does A differ from B, > >>>>> C, IN, or OUT? > >>>>> > >>>>> Stephen > >>>> Do 1+n, 2*n and sin(n*pi) look like formulas to you? They do to me. > >>>> Maybe they're just the names of your cats? > >>> Sure they are formulas. But I am interested in your phrase > >>> "formulaic relationship", the explanation of which you seem to be avoiding. > >>> > > > >> It's the mapping between set using a quantitative formula. Observe... > > > >>>> A can be expressed 1+n>=1, or n>=0, and is the set mapped from the > >>>> naturals neN (starting from 1) by the formula f(n)=n-1. The inverse of > >>>> n-1 is n+1, indicating that over all values, this set has one more > >>>> element than N, namely, 0. > >>> I said that n was restricted to non-negative integers, so this > >>> set equals N. > >>> > > > >> Ooops, missed that. Sorry. n is restricted to nonnegative integers, but > >> f(n) isn't. What you mean is that, in this case, f(n) is restricted to > >> nonnegative integers, which means n>=2, and f(n)>=1. So, yes, the set is > >> size N, from 1 through N. > > > >>>> B can be simplified by subtracting n from both sides, without any worry > >>>> of changing the inequality, so we get n>=0, neN. That's the same set, > >>>> again, mapped from the naturals by f(n)=n-1. > >>> Also N. > > > >> Yes, by the same reasoning. > > > >>>> C is simply the set of all integers, which we can consider twice the > >>>> size of N. There's really nothing to formulate about that. > >>> Once again N. > >>> > > > >> Sure. > > > >>> So all three sets are N. So in fact, there is only one set. > >>> A, B, and C are all the same set. A, B, C, IN and OUT are all > >>> the same set, namely N. You still have not answered what > >>> a "formulaic relationship" is. > >>> > >>> Stephen > > > >> Take the set of evens. It's mapped from the naturals by f(x)=2x. Right. > >> Many feel that there are half as many evens as naturals, and this is > >> reflected in the inverse of the mapping formula, g(x)=x/2. Over the > >> range of N, we have N/2 as many evens as naturals. Over the range of N, > >> we have sqrt(N) as many squares as naturals, and log2(N) as many powers > >> of 2 in N. That's IFR, using formulaic relationships between infinite > >> sets. Byt he way, it works for finite sets, too. :) > > > > What does that have to do with the sets IN and OUT? IN and OUT are > > the same set. You claimed I was losing the "formulaic relationship" > > between the sets. So I still do not know what you meant by that > > statement. Once again > > IN = { n | -1/(2^(floor(n/10))) < 0 } > > OUT = { n | -1/(2^n) < 0 } > > > > I mean the formula relating the number In to the number OUT for any n. You've lost a capital 'N'; but anyway - IN and OUT are sets. They are not numbers. Yes, they are sets of numbers, but in mathematics numbers and sets of numbers are different things. > That is given by out(in) = in/10. Well, you've lost a capital 'I' now. Or is "in" supposed to be something else? Look, the two sets above are "produced" at exactly the same "rate" (insofar as I speak poetry). For each natural number n, we check the condition -1/(2^(floor(n/10))) < 0 to determine if it belongs to IN, and the condition -1/(2^n) < 0 to see if it belongs to OUT. For all n greater than zero, it turns out that both conditions are true, and therefore each of these positive naturals is popped into IN and popped into OUT. Simultaneously (poetically speaking). > The formulaic relationship is lost in that statement. When you state the > relationship given any n, then the answer is obvious. Do "state the relationship given any n"... I mean, what is it, exactly? Brian Chandler http://imaginatorium.org
From: Tony Orlow on 28 Oct 2006 14:50 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: >>> Tony Orlow wrote: >>>> Virgil wrote: >>>>> In article <454286e8(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> stephen(a)nomail.com wrote: >>>>>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>>>>>>> Tony Orlow wrote: >>>>>>>>> David Marcus wrote: >>>>>>>>>> Your question "Is there a smallest infinite number?" lacks context. You >>>>>>>>>> need to state what "numbers" you are considering. Lots of things can be >>>>>>>>>> constructed/defined that people refer to as "numbers". However, these >>>>>>>>>> "numbers" differ in many details. If you assume that all subjects that >>>>>>>>>> use the word "number" are talking about the same thing, then it is >>>>>>>>>> hardly surprising that you would become confused. >>>>>>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm >>>>>>>>> not interested in that nonsense, to be honest. I see it as a dead end. >>>>>>>>> >>>>>>>>> If there is a definition for "number" in general, and for "infinite", >>>>>>>>> then there cannot both be a smallest infinite number and not be. >>>>>>>> A moot point, since there is no definition for "'number' in general", as >>>>>>>> I just said. >>>>>>>> -- >>>>>>>> David Marcus >>>>>>> A very simple example is that there exists a smallest positive >>>>>>> non-zero integer, but there does not exist a smallest positive >>>>>>> non-zero real. If someone were to ask "does there exist a smallest >>>>>>> positive non-zero number?", the answer depends on what sort >>>>>>> of "numbers" you are talking about. >>>>>>> >>>>>>> Stephen >>>>>> Like, perhaps, the Finlayson Numbers? :) >>>>> Any set of numbers whose properties are known. Are the properties of >>>>> "Finlayson Numbers" known to anyone except Ross himself? >>>> Uh, yeah, I think I understand what his numbers are. Perhaps you've seen >>>> our recent exchange on the matter? They are discrete infinitesimals such >>>> that the sequence of them within the unit interval maps to the naturals >>>> or integers on the real line. Is that about right, Ross? >>> Do they form a field? >>> >>> Brian Chandler >>> http://imaginatorium.org >>> >> Good question. Ross? What says you to this? > > I'm agog to hear Ross's answer... > >> Here's what Wolfram says applies to fields: >> http://mathworld.wolfram.com/FieldAxioms.html >> >> My understanding, looking at each of these axioms, is that they apply to >> this system, and that it's a field. I suppose you would want proof of >> each such fact, but perhaps you could move the process along by >> suggesting which of the ten axioms you think the Finlayson Numbers might >> violate? After all, if you find only one, then you've proved your point. > > Well, isn't 'iota' the "smallest positive Freal"? And isn't 2 also a > Freal? So what is iota/2? > That's a good question. In T-riffics, it's 0.000...000.5 (decimal) or 0.000...0001 (binary). > >> Not that I am necessarily concerned with whether they form a ring or a >> field or whatever, until that becomes important. Is it? Why the question? > > No, no, concentrate on the poetry. "Why the question?" you ask - some > might think it was just to make you look silly and ignorant, but in > fact that's not necessary at all. > > Brian Chandler > http://imaginatorium.org > Oh. So, that's why. Nice to know....
From: Tony Orlow on 28 Oct 2006 14:52
Lester Zick wrote: > On 27 Oct 2006 11:38:10 -0700, imaginatorium(a)despammed.com wrote: > >> David Marcus wrote: >>> Lester Zick wrote: >>>> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: >>>>> A very simple example is that there exists a smallest positive >>>>> non-zero integer, but there does not exist a smallest positive >>>>> non-zero real. >>>> So non zero integers are not real? >>> That's a pretty impressive leap of illogic. >> Gosh, you obviously haven't seen Lester when he's in full swing. (Have >> _you_ searched sci.math for "Zick transcendental"?) > > Hell, Brian, on some of my better days I can even prove the pope's > catholic. > > ~v~~ That must be a proof by contradiction. It doesn't involve a largest finite, does it? ;) 01oo |